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- W112093935 abstract "In Chap. 1 we considered Markov chains Xn with a discrete time index n = 0, 1, 2, … In this chapter we will extend the notion to a continuous time parameter t ≥ 0, a setting that is more convenient for some applications. In discrete time we formulated the Markov property as: for any possible values of j, i, in−1, … i0$$displaystyle{P(X_{n+1} = jvert X_{n} = i,X_{n-1} = i_{n-1},ldots,X_{0} = i_{0}) = P(X_{n+1} = jvert X_{n} = i)}$$In continuous time, it is technically difficult to define the conditional probability given all of the Xr for r ≤ s, so we instead say that Xt, t ≥ 0 is a Markov chain if for any 0 ≤ s0 < s1⋯ < sn < s and possible states i0, …, in, i, j we have $$displaystyle{P(X_{t+s} = jvert X_{s} = i,X_{s_{n}} = i_{n},ldots,X_{s_{0}} = i_{0}) = P(X_{t} = jvert X_{0} = i)}$$In words, given the present state, the rest of the past is irrelevant for predicting the future. Note that built into the definition is the fact that the probability of going from i at time s to j at time s + t only depends on t the difference in the times." @default.
- W112093935 created "2016-06-24" @default.
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- W112093935 date "2012-01-01" @default.
- W112093935 modified "2023-09-26" @default.
- W112093935 title "Continuous Time Markov Chains" @default.
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- W112093935 doi "https://doi.org/10.1007/978-1-4614-3615-7_4" @default.
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